We'll start by discussing the formal error bound for Taylor polynomials. (I.e., how badly does a Taylor polynomial approximate a function?) Then, we'll see some standard examples. Finally, we'll see a powerful application of the error bound formula.

Lagrange Error Bound for

We know that the th Taylor polynomial is , and we have spent a lot of time in this chapter calculating Taylor polynomials and Taylor Series. The question is, for a specific value of , how badly does a Taylor polynomial represent its function? We define the error of the th Taylor polynomial to be

That is, error is the actual value minus the Taylor polynomial's value. Of course, this could be positive or negative. So, we force it to be positive by taking an absolute value.

The following theorem tells us how to bound this error. That is, it tells us how closely the Taylor polynomial approximates the function. Essentially, the difference between the Taylor polynomial and the original function is at most . At first, this formula may seem confusing. I'll give the formula, then explain it formally, then do some examples. You may want to simply skip to the examples.

Theorem 10.1 Lagrange Error Bound Let be a function such that it and all of its derivatives are continuous. If is the th Taylor polynomial for centered at , then the error is bounded by

where is some value satisfying on the interval between and .

Explanation

We derived this in class. The derivation is located in the textbook just prior to Theorem 10.1. The main idea is this: You did linear approximations in first semester calculus. What you did was you created a linear function (a line) approximating a function by taking two things into consideration: The value of the function at a point, and the value of the derivative at the same point. You built both of those values into the linear approximation. A Taylor polynomial takes more into consideration. It considers all the way up to the th derivative. So, the first place where your original function and the Taylor polynomial differ is in the st derivative. Really, all we're doing is using this fact in a very obscure way. We differentiated times, then figured out how much the function and Taylor polynomial differ, then integrated that difference all the way back times.

Basic Examples

Find the error bound for the rd Taylor polynomial of centered at on .

Solution: This is really just asking “How badly does the rd Taylor polynomial to approximate on the interval ?” Intuitively, we'd expect the Taylor polynomial to be a better approximation near where it is centered, i.e. near . We have

where bounds on the given interval . But, we know that the 4th derivative of is , and this has a maximum value of on the interval . So, we have . Thus, we have a bound

given as a function of . Now, if we're looking for the worst possible value that this error can be on the given interval (this is usually what we're interested in finding) then we find the maximum value that can take on the given interval. That maximum value is . Hence, we know that the 3rd Taylor polynomial for is at least within

of the actual value of on the interval .

What is the maximum possible error of the th Taylor polynomial of centered at on the interval ?

Solution: We have

where bounds on . Since takes its maximum value on at , we have . Thus, we have

What is the worst case scenario? When is the largest is when . Thus, we have

In other words, the 100th Taylor polynomial for approximates very well on the interval .

A More Interesting Example

Problem: Show that the Taylor series for is actually equal to for all real numbers .

Proof: The Taylor series is the “infinite degree” Taylor polynomial. So, we consider the limit of the error bounds for as . That is, we're looking at

Since all of the derivatives of satisfy , we know that . Thus, we have

But, it's an off-the-wall fact that

Thus, we have shown that

for all real numbers . Thus, as , the Taylor polynomial approximations to get better and better.